Wronskian

How can we be sure that our general set of solutions is indeed the general set of solutions? The Wronskian!

Verifying Some of Our General Solutions
In the section "Characteristic Equation with Repeated Roots" we had the following differential equation:
$y''-6y'+9y=0$
It was found that the two solutions were:
$y_1 (x)=e^{3x}$ $y_2 (x)=xe^{3x}$
1. But one might have made the assumption that both solutions were of the form: $y_1 (x)=y_2 (x)=e^{3x}$ . Demonstrate that these solutions are not linearly independent
2. Show that the two actual solutions actually form a fundamental set of solutions.
In the section "Characteristic Equation with Real Distinct Roots" we had the following differential equation:
$6y''+8y'-8y=0$ We found two solutions:

$y_1 (x)=e^{\frac{2}{3} x}$
$y_2 (x)=e^{-2x}$

Verify that these two solutions are indeed linearly independent
A certain differential equation was found to have two solutions:
$y_1(x)=3 \cos (2x)$ $y_2 (x)=3-6\sin^2 (x)$ Are these two solutions independent? Will they form a fundamental set of solutions?